New Algorithms for Implementing and Interpolating Rotations

We have not found in the computer graphics literature what we believe may be the simplest algorithm for implementing a 3D rotation with a mouse. We present this algorithm, and observe that it leads to new mathematical realizations of the 3D rotation group and its double-cover, the unit quaternions. We use this to obtain an algorithm for interpolating rotations of a 3D scene that is more efficient than the standard quaternion based methods, and also to derive the quaternion composition formula from the geometry of 3D rotations. 1. Implementing prescribed rotations by reflecting pairs. There are many methods well-known in computer graphics for implementing a three-dimensional rotation R ∈ SO(3) specified by two-dimensional user input from various controllers, e.g., a joystick, a trackball, or a mouse (e.g., [5–7, 22, 40– 41]). These algorithms are typically based on a dynamic mapping of the position of the controller to a unit vector in R (we describe below the procedure for the case of a mouse used as a virtual trackball). Such a mapping converts the initial position of the controller to a unit vector uI ∈ R, and when that position is moved, the final position is converted to a second unit vector uF ∈ R. For the object on the screen to follow the mouse naturally, as if it were being dragged, we must perform a rotation ρ on it satisfying uF := ρuI . But there are infinitely many such rotations, since we could also perform any rotation about uI , followed by ρ, followed by any rotation about uF , and still have uI map to uF . However, if uF 6= −uI , there is a unique rotation that takes uI to uF and that acts as the identity on the orthogonal complement of V := span{uI ,uF }. It is this rotation that is conventionally specified by giving a unit vector and its image, and we call it the transvection ([4, 21, 49]) taking uI to uF , denoted T(uI ,uF ). Note that, by orthogonality, T(uI ,uF ) preserves V . In the computer graphics literature, a number of different methods have been recommended for implementing T(uI ,uF ), including: construction of the rotation matrix (Chen, [5–7, 28)), using axis-angle (Euler-Rodrigues) formulas ([14, 26, 31, 32, 35]), construction of a unit quaternion, Q, and then conjugating by Q or else converting Q to a matrix (Shoemake, [40–41]), and even using Euler angles. Similar needs for specifying, implementing, and composing rotations, and proposed solutions in terms of quaternions and matrices also arise in other fields, including robotics ([11, 27, 37, 38]), photogrammetry ([25]), and rocket motion control ([45]).